Expected Returns and Expected Growth in Rents of Commercial Real Estate

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1 Expected Returns and Expected Growth in Rents of Commercial Real Estate Alberto Plazzi Walter Torous Rossen Valkanov This Version: January 2010 Abstract Commercial real estate expected returns and expected rent growth rates are timevarying. Relying on transactions data from a cross-section of U.S. metropolitan areas, we find that up to 30% of the variability of realized returns to commercial real estate can be accounted for by expected return variability, while expected rent growth rate variability explains up to 45% of the variability of realized rent growth rates. The cap rate, that is, the rent-price ratio in commercial real estate, captures fluctuations in expected returns for apartments, retail, as well as industrial properties. For offices, by contrast, cap rates do not forecast (in-sample) returns even though expected returns on offices are also time-varying. As implied by the present value relation, cap rates marginally forecast office rent growth but not rent growth of apartments, retail and industrial properties. We link these differences in in-sample predictability to differences in the stochastic properties of the underlying commercial real estate data generating processes. Also, rent growth predictability is mostly observed in locations characterized by higher population density and stringent land use restrictions. The opposite is true for return predictability. The dynamic portfolio implications of time-varying commercial real estate returns are also explored in the context of a portfolio manager investing in the aggregate stock market, Treasury bills as well as commercial real estate. We thank Andrea Berardi, Markus Brunnermeier (AFA discussant), Christopher Downing, Erasmo Giambona (EFA discussant), Harrison Hong, Andrey Pavlov, Antonio Rubia, Stephen Schaefer and seminar participants at the following conferences for useful comments: the AFA 2007 Meeting, the AREUEA, the XXXIV EFA Meeting, the XXXVI EWGFM, the FMA European meeting, the SAET meeting in Vigo, and the SAFE center at the University of Verona. We are especially grateful to an anonymous referee and the Editor, Matthew Spiegel, for many suggestions that have greatly improved the paper. All remaining errors are our own. The Anderson School at UCLA and SAFE Center University of Verona, 110 Westwood Plaza, Los Angeles, CA , phone: (310) , fax: (310) , The Anderson School at UCLA, 110 Westwood Plaza, Los Angeles, CA , phone: (310) , fax: (310) , Corresponding author. Rady School at UCSD, Pepper Canyon Hall, 9500 Gilman Drive, MC 0093, La Jolla, CA 92093, phone: (858) , fax: (858) , The latest draft is available at:

2 1 Introduction U.S. commercial real estate prices fluctuate considerably, both cross-sectionally as well as over time. For example, the return to apartment buildings during the last quarter of 1994 ranged from 21.4 percent in Dallas, Texas to 8.5 percent in Portland, Oregon. Eight years later, during the last quarter of 2002, the returns to apartments in Dallas and Portland were 1.2 percent and 4.4 percent, respectively. Other types of commercial real estate, such as retail, industrial, and office properties, have experienced even larger return fluctuations. Understanding what drives these fluctuations is an important research question as commercial real estate represents a substantial fraction of total U.S. wealth. For example, Standard and Poor s estimates the value in 2007 of all U.S. commercial real estate to be about $5.3 trillion, about one fifth of the stock market s value. From an asset pricing perspective, the price of a commercial property, be it an office building, apartment, retail or industrial space, equals the present value of its future rents. This fundamental present value relation implies that observed fluctuations in commercial real estate prices should reflect variation in future rents, or in future discount rates, or both. The possibility of time-varying discount rates and rent growth rates should be explicitly considered in the valuation of commercial real estate as it is often conjectured that both fluctuate with the prevailing state of the economy. Case (2000), for example, points out the vulnerability of commercial real estate values to changes in economic conditions by describing recent boom-and-bust cycles in that market. He provides a simple example of cyclical fluctuations in expected returns and rent growth rates that give rise to sizable variation in commercial real estate values. Case, Goetzmann, and Rouwenhorst (2000) make a similar point using international data and conclude that commercial real estate is a bet on fundamental economic variables. Despite these and other studies, little is known about the dynamics of commercial real estate returns and rent growth rates. In this paper, we investigate whether expected returns and expected growth in rents of commercial real estate are time-varying by relying on a version of Campbell and Shiller s (1988b) dynamic Gordon model. An implication of this model is that the cap rate, defined as the ratio between a property s rent and its price, should reflect fluctuations in expected returns, or in rent growth rates, or both. The cap rate is a standard measure of commercial real estate valuation and corresponds to a common stock s dividend-price ratio where the property s rent plays the role of the dividend. By way of example, suppose that the cap rate for apartment buildings in Portland is higher than the cap rate of similar properties in Dallas. The dynamic Gordon model implies that either future discount rates in Portland will be higher than those expected in Dallas, or that future rents in Portland are expected to grow at a slower rate than in Dallas, or both. Commercial real estate offers a number of advantages over common stock when investigating the dynamics of expected returns and the growth in cash payouts. For example, it is often argued 1

3 that common stock dividends do not accurately reflect the changing investment opportunities confronting a firm. Dividends are paid at the discretion of the firm s management and there is extensive evidence that they are either actively smoothed, the product of managers catering to particular clienteles, or the result of management s reaction to perceived mispricing (Shefrin and Statman (1984), Stein (1996), and Baker and Wurgler (2004)). By contrast, rents on commercial properties are not discretionary and are paid by tenants as opposed to property managers. Furthermore, commercial rents are particularly sensitive to prevailing economic conditions such as employment and growth in industrial production (DiPasquale and Wheaton (1996)). In conducting our empirical analysis, we rely on a novel data set summarizing commercial real estate transactions across fifty-three U.S. metropolitan areas reported at a quarterly frequency over a sample period extending from the second quarter of 1994 to the first quarter of For a subset of twenty-one of these areas, we also have bi-annual observations beginning in the last quarter of These data are available for a variety of property types including offices, apartments, as well as retail and industrial properties. The transactions nature of our commercial real estate data differentiates it from the appraisal data typically relied upon in other studies. For example, unlike the serially correlated returns and rent growth rates characterizing appraisal data (Case and Shiller (1989, 1990)), returns and rent growth rates in our data are not serially correlated beyond a yearly horizon. We find that higher cap rates predict higher future returns of apartment buildings as well as retail and industrial properties. Cap rates, however, do not predict the future returns of office buildings. For apartments, retail, and industrial properties, the predictability of returns is robust to controlling for cross-sectional differences using variables that capture regional variation in demographic, geographic, and various economic factors. In terms of the economic significance of this relation, we find that a one percent increase in cap rates leads to an increase of up to four percent in the prices of these properties. This large effect is due to the persistence of the fluctuations in expected returns and is similar in magnitude to that documented for common stock (Cochrane (2008)). The evidence of return predictability is primarily drawn from locations characterized by lower population density and less land use restrictions. By contrast, we do not find reliable evidence that cap rate fluctuations are associated with future movements in rent growth rates. Only for offices do we find some evidence of higher cap rates predicting lower future rent growth rates and then only at long horizons. Also, rent growth predictability is more likely to be observed in locations characterized by higher population density and stricter land use restrictions. These findings, however, should only be interpreted as in-sample evidence because our estimation procedure relies on data drawn from the entire sample period. Therefore, our results are not predictive in the sense that an investor in real-time would have been able to replicate them or profit from them (Goyal and Welch (2008)). 2

4 Taken together, our findings point to a fundamental difference between apartments, retail and industrial properties, where cap rates do forecast returns, versus office buildings, where they do not. This difference provides us with a unique opportunity to investigate under what circumstances valuation ratios of broadly similar assets commercial real estate properties can or cannot predict returns and the growth in cash payouts. To do so, we formulate an underlying structural model for returns and rent growth whose conditional expectations vary over time. An important feature of the model is that it captures the co-movement between the time-varying components of expected returns and expected rent growth ((Menzly, Santos, and Veronesi (2004), Lettau and Van Nieuwerburgh (2008), and Koijen and Van Binsbergen (2009)). Such a co-movement is not only supported by our data but, as we demonstrate, also generates systematic patterns in the coefficients of predictive regressions. Relying on an identification scheme which makes use of the estimated predictive regression coefficients as well as other moments of the data, we explicitly relate the predictive regression coefficients to the stochastic properties of returns and rent growth. By doing so, we are able to explain observed differences in the forecasting ability of cap rates across property types in terms of the underlying data generating processes. The ability of the cap rate to forecast returns or rent growth depends primarily on two factors: the co-movement between expected returns and expected rent growth, and the extent to which variation in expected rent growth is orthogonal to the time-varying component of expected returns. Interestingly, the co-movement factor has a non-monotonic effect on the predictive regressions slope coefficients. This reflects the fact that a greater co-movement reduces both the variability of the cap rate as well as its covariance with future returns. Variation in expected rent growth orthogonal to the time-varying component of expected returns represents noise in the cap rate s ability to forecast returns. If cap rates are driven primarily by this orthogonal component then the cap rate s ability to forecast returns will be reduced while, at the same time, its ability to forecast rent growth will be improved. Because the cap rate is correlated with both expected returns and expected rent growth, this predictor is imperfect as in Pastor and Stambaugh (2009). However, unlike their reduced-form setup where expected returns are imperfectly correlated with a predictor, we follow Lettau and Van Nieuwerburgh (2008) and explicitly link the structural parameters of the data generating process to the coefficients of the predictive regression. We find that offices are characterized by the largest co-movement between expected returns and expected rent growth and also have a relatively large noise component when compared to the other property types. These two features imply that office cap rates are an imperfect and extremely noisy proxy of future returns. This lack of predictability, however, does not mean that the expected return to offices is not time varying. To the contrary, we document that commercial real estate expected returns and expected rent growth are extremely volatile across all property types. In particular, the variability of realized returns accounted for by the variability of expected 3

5 returns over the 1985 to 2002 sample period range from approximately 16% in the case of industrial properties to almost 30% in the case of retail properties. For rent growth rates over this same sample period, approximately 22% of the variability of realized industrial rent growth rates is explained by the variability of expected industrial rent growth rates, while, at the other extreme, expected office rent growth rates explain almost 43% of the variability of realized office rent growth rates. Our empirical methodology differs in a number of ways from the standard approach of estimating predictive regressions by ordinary least squares (OLS). In particular, we rely on a generalized method of moments (GMM) procedure in which the present value constraint is imposed to jointly estimate the returns and rent growth predictive regressions. This restriction renders the system internally consistent as these regressions are explicitly related by the Campbell and Shiller (1988b) log-linearization. The predictive relations at all horizons are then estimated simultaneously as a system as opposed to horizon by horizon. We also use the entire cross-section of metropolitan areas when estimating the system for each commercial property type. Combining cross-sectional and time-series information improves the efficiency with which the predictive relations will be estimated. Finally, we rely on a double-resampling procedure to take into account the overlapping and cross-sectional nature of our data. The view of commercial real estate that emerges from our analysis is that of an asset class characterized by fluctuations in expected returns not unlike that of common stock. This being the case, a natural question to ask is whether, like common stock, the allocation of commercial real estate within a portfolio can be improved by exploiting the time variation of its expected return. To do so, we use the parametric portfolio approach of Brandt and Santa-Clara (2006) to investigate the properties of a dynamic portfolio strategy in which funds are allocated between Treasury bills, common stock, and commercial real estate by taking advantage of the information provided by the dividend-price ratio as well as the cap rate. Including the cap rate in addition to the dividend yield has a significant impact on the dynamic asset allocation. In particular, the resultant dynamic portfolio strategy results in an increase of the Sharpe ratio from approximately 0.5 to almost 1.0 in comparison to the corresponding static portfolio position (which ignores predictability) and yields a certainty equivalent return of over 6 percent per year. The plan of this paper is as follows. In Section 2, we present our commercial real estate valuation framework and the corresponding predictive regressions used to investigate the dynamics of commercial real estate returns and rent growth. The time-series and cross-sectional properties of our transactions-based commercial real estate data are also presented. These properties motivate an underlying structural model of the commercial real estate data generating process. The predictive regression coefficients can then be interpreted in terms of the properties of commercial real estate returns and rent growth implied by the structural model. Section 3 details a two-step estimation procedure which takes advantage of the pooled nature of our data to efficiently estimate predictive 4

6 regressions. The results of estimating in-sample the predictive regressions and the corresponding structural parameters are then presented. In section 4, we investigate cross-sectional differences in our predictive regression results based on population density and land use regulation. We also test the robustness of our results to the inclusion of other cross-sectional determinants of commercial real estate returns and rent growth across metropolitan areas. Section 5 investigates the properties of a dynamic portfolio strategy which exploits the predictability of commercial real estate returns as well as the predictability of common stock returns. We offer concluding remarks in Section 6. 2 Commercial Real Estate Returns and Rents 2.1 The Cap Rate Model We denote by P m i,t the price of a commercial property in area i at the end of period t where the superscript m refers to the type of property being considered: apartments, office buildings, industrial, or retail properties. 1 Similarly, H m i,t+1 is the net rent2 of a commercial property of type m in area i from period t to t+1. The gross return from holding a given commercial property from t to t + 1 is defined by: 1 + R m i,t+1 P i,t+1 m + Hm i,t+1 Pi,t m. (1) The definition of the return to commercial real estate is similar to that of common stock. The only difference is that a commercial property pays rental income instead of a dividend. The rent-to-price ratio Hi,t m/p i,t m, known as the cap rate in the commercial real estate industry (Geltner and Miller (2000), Brueggeman and Fisher (2008)), corresponds to the dividend-price ratio for stocks. We define the log price, log return, log rent, and log rent-to-price ratio as p m i,t log(p m i,t ), ri,t+1 m log(1 + Rm i,t+1 ), hm i,t+1 log(hm i,t+1 ), and capm i,t h m i,t pm i,t, respectively. We follow Campbell and Shiller (1988b) and express ri,t+1 m using a first-order Taylor approximation as ri,t+1 m κ+ρpm i,t+1 +(1 ρ)hm i,t+1 pm i,t, where κ and ρ are parameters derived from the linearization.3 Solving this relation forward, imposing the transversality condition lim k ρ k p m i,t+k = 0 to avoid the presence of rational bubbles, and taking expectations at time t, we can write the following 1 Hotel properties represent another category of commercial real estate. Unfortunately, we do not have data on hotels and so they are excluded from our subsequent analysis. However, hotels represent less than four percent of the total value of U.S. commercial real estate. See Case (2000). 2 Net rents require the tenant, as opposed to the landlord, to be responsible for operating expenses such as electricity, heat, water, maintenance, and security (see Geltner and Miller (2000)). When there is no possibility of confusion, we will refer to net rents simply as rents. 3 In particular, ρ 1/(1 + exp(h p)) where h p denotes the average log rent - price ratio. Note that for the US commercial real estate market, the average rent - price ratio across property types and metropolitan areas for the sample period is approximately 9% on an annual basis, implying an annualized value for ρ of This is lower than the 0.98 annualized value of ρ for the aggregate stock market during the same period. 5

7 expression for the log cap rate: [ cap m i,t = κ ] [ 1 ρ + E ] t ρ k ri,t+1+k m E t ρ k h m i,t+1+k. (2) k=0 It should be noted that this log linearization is valid only if expected returns and rent growth rates are both stationary. Whether or not this is the case in our commercial real estate data will be investigated later. The preceding expression for the cap rate is best understood as a consistency relation. If a commercial property s cap rate is high, then either the property s expected return is high, or the growth of its rents is expected to be low, or both. This log-linearization framework was proposed by Campbell and Shiller as a generalization of Gordon s (1962) constant-growth model and explicitly allows both expected returns and dividend growth rates to be time-varying. Like other assets, there are good reasons to believe that expected returns to commercial real estate and rent growth rates are both time-varying. We use expression (2) as the starting point for our analysis of fluctuations in commercial real estate prices. 4 For a particular property type in a specific area, expression (2) states that the cap rate forecasts either expected returns or expected growth in rents, or both. This is usually tested by estimating the following system: k=0 r t+1 = α + β (cap t ) + ε r t+1 (3) h t+1 = µ + λ (cap t ) + ε h t+1 (4) cap t+1 c = φ (cap t c) + ε c t+1. (5) Expressions (3) and (4) are predictive regressions which relate future returns and rent growth, respectively, to today s cap rate while expression (5) describes the dynamics of the cap rate. We collect the residuals of these regressions in the vector ε t+1 = [ε r t+1, εh t+1, εc t+1 ] and denote its variance-covariance matrix by Σ ε. Predictive regressions have been extensively analyzed in the context of the stock market by, among others, Campbell and Shiller (1988b), Fama and French (1988), Cochrane (2008), and Lettau and Van Nieuwerburgh (2008). As emphasized by this literature, absent any further assumptions, the residuals ε t+1 have no clear economic interpretation and are simply forecasting or measurement errors. In the same spirit, expressions (3)-(5) are best interpreted as the reduced form of an underlying data generating process that is usually left unspecified. Like much of the predictability literature, we estimate these regressions in-sample. That is, 4 To the extent that this expression holds for any property type in any particular area, without loss of generality, we will simplify our subsequent notation by simply keeping track of the time subscript. 6

8 our estimates of the parameters of these predictive regressions are based on the entire sample and not just the data available at the time a particular cap rate observation is realized. Therefore, as emphasized by Goyal and Welch (2008), these results be taken as documenting ex-post variation in conditional means rather than an ex-ante forecasting relation. An important insight of the predictability literature is that the forecasting ability of a slowly moving predictor may be more easily discerned at long horizons as opposed to short horizons. That being the case, we will test whether cap rates forecast future returns and rent growth over a horizon of k > 1 periods: r i,t+1 t+k = α k + β k (cap t ) + ε r t+1 t+k (6) h i,t+1 t+k = µ k + λ k (cap t ) + ε h t+1 t+k (7) where r t+1 t+k k i=0 r t+1+i and h t+1 t+k k i=0 h t+1+i proxy for expected returns and expected rent growth rates, respectively. Under the assumption that the cap rate follows an AR(1) process with autoregressive coefficient φ, as described by expression (5), the short-horizon and corresponding long-horizon coefficients of the predictive regressions 5 are related by: ( ) 1 φ k β k = β 1 φ and ( ) 1 φ k λ k = λ. (8) 1 φ We will be able to gain further economic insights into the results of these predictive regressions if additional structure is imposed on the dynamics of returns and rent growth rates. To do so requires that we first introduce our commercial real estate data and describe its time-series and cross-sectional properties. 2.2 Commercial Real Estate Data Our commercial real estate data consists of prices, Pi,t m, and cap rates, CAP i,t m, of class A offices, apartments, retail and industrial properties located in fifty-three U.S. metropolitan areas and are provided by Global Real Analytics (GRA). The prices and cap rates for each property type in a particular area are value-weighted averages of corresponding transactions data in a given quarter. 6 Class A buildings are investment-grade properties that command the highest rents and sales prices in a particular market. 7 The fifty-three sampled areas encompass more than 60% of the U.S. population. A listing of these metropolitan areas is given in Appendix A. The data are available 5 In what follows, the coefficients β and λ always refer to the corresponding one-period regression coefficients, while the subscript k applies to long-horizon regression coefficients. 6 GRA will not disclose details surrounding the construction of their data series apart from the fact that all averages for a particular property type are value-weighted and are based on at least twelve transactions within a metropolitan area in a given quarter. Cap rates are calculated based on net rents. 7 Class B buildings, by contrast, are less appealing and are generally deficient in floor plans, condition and facilities. Class C buildings are older buildings that offer basic services and rely on lower rents to attract tenants. 7

9 on a quarterly basis beginning in the second quarter of 1994 (1994:Q2) and ending in the first quarter of 2003 (2003:Q1). Taken together, we have panel data consisting of 1908 observations (36 quarters 53 metropolitan areas). 8 We use the given P m i,t and CAP m i,t to construct quarter t s net rents as Hm i,t = (CAP m i,t P m i,t )/4.9 The gross returns 1+R m i,t in quarter t are then obtained from expression (1) while Hm i,t /Hm i,t 1 gives one plus the growth in rents. For consistency, we work with log cap rates, cap m i,t = ln(cap m i,t ), and log rent growth rates, h m i,t = ln(hm i,t /Hm i,t 1 ). We also rely on log excess returns, rm i,t = ln(1 + R m i,t ) ln(1 + TBL t), where TBL denotes the three month Treasury bill yield. 10 Our returns and rent growth series are based exclusively on transactions data and are available across a cross-section of metropolitan areas. By concentrating only on class A buildings, GRA attempts to hold property quality constant both within a particular metropolitan area as well as across metropolitan areas. 11 While these particular indices are no longer publicly available, they serve as the basis of GRA s recent efforts in conjunction with Standard & Poor s to offer a series of commercial real estate indices for the trading of commercial real estate futures and options on futures on the Chicago Mercantile Exchange. Table 1 presents summary statistics of the key variables in our analysis - excess returns, rent growth rates, and cap rates - during the 1994:Q2 to 2003:Q1 sample period. We report time-series averages, standard deviations, and serial correlations at both one-quarter and one-year lags. We also report t-statistics testing the null hypotheses of no serial correlation in the excess returns and rent growth series as well as the augmented Dickey-Fuller (ADF) statistic testing the null hypothesis of a unit root in the cap rate series. In the interest of brevity, we report only the averages of these statistics across the fifty-three sampled metropolitan areas for each of the four commercial property types. From Table 1 we see that the average annualized excess returns of the commercial properties range from 7.3% (offices) to 9.5% (apartments) while the average annualized standard deviations lie between 3.7% (retail) and 6.1% (apartments). By comparison, the corresponding average annualized return of the CRSP-Ziman REIT value-weighted index is comparable at 8.8% but with a standard deviation of 12.4%. The higher volatility of the REIT index reflects the fact 8 A subset of these data are available for twenty-one of the metropolitan areas going back to 1985:Q4 and ending in 2002:Q4 but only at a bi-annual frequency. 9 We obtain very similar results by modifying the timing convention and relying on the expression Hi,t m = (CAPi,t m Pi,t 1)/4. m 10 We use excess returns throughout the paper, since our main interest is in variation in risk premia. When we use real rather than excess returns, the results are very similar (see also Cochrane (2008)). For the sake of brevity, excess returns will be referred to simply as returns. 11 Commercial real estate turns over far less frequently than residential real estate. This extremely low turnover rate, especially in certain metropolitan areas, makes it difficult to construct a repeat-sales index for commercial properties. 8

10 that REITs are leveraged investments that are traded much more frequently than commercial real estate properties. 12 The commercial property excess return series have low serial correlations at a one-quarter lag and virtually none at a yearly lag. For each property type, the average serial correlation at a one-quarter lag is not statistically significantly different from zero. The number of significant one-quarter autocorrelations for any particular property type located in a given metropolitan area (not displayed) is small. Only for offices is the average serial correlation at a one-quarter lag relatively high but it remains statistically indistinguishable from zero. At an annual lag, the excess return series are, on average, close to uncorrelated for all property types. Similarly, very few one-year autocorrelations for a particular property type located in a given metropolitan area are statistically significant (not displayed). While serial correlation tests may have little power, the general message that emerges is that these excess returns do not appear to suffer from the high autocorrelations which plague appraisal-based series. 13 The autocorrelation properties of the rent growth series are similar, exhibiting modest serial correlations at both one-quarter and one-year lags. The lack of persistence in the excess returns and rent growth series can also be seen in the respective values of the t-statistics given in the final column of Table 1 which indicate that we cannot reject the null hypothesis of zero first-order autocorrelations in either series. Turning our attention to cap rates, the annualized average cap rates in Table 1 range from 8.9% (apartments) to 9.3% (retail) and lie between the estimates of Liu and Mei (1994), who report average cap rates for commercial real estate of approximately 10.4%, and Downing, Stanton, and Wallace (2008) s range of 7.8% to 8.5%. 14 In contrast to the excess return series, however, the cap rate series are extremely persistent. The average first-order serial correlation at a quarterly lag lies between (industrial) and (offices) and we obtain serial correlation estimates close to unity for commercial properties located in many metropolitan areas (not displayed). The final column of Table 1 reports the average ADF statistic under the null hypothesis of a unit root in the cap rate series. As can be seen, this null hypothesis cannot be rejected for any of the property types, though we also cannot reject local-to-unity alternatives such as φ = The time-series properties of the cap rates have important implications for our subsequent 12 During the same sample period, the annualized return of the CRSP value-weighted stock market index had a mean and standard deviation of 8.4% and 19.2%, respectively, while the annualized three-month Treasury bill rate had a mean and standard deviation of 4.5% and 0.7%, respectively. 13 See Geltner (1991) for a detailed discussion of this issue. 14 The average dividend-price ratio for the CRSP-Ziman REIT Value-Weighted Index over this sample period is 6.8%. 15 These φ estimates also suffer from a downward bias (Andrews (1993)) which our subsequent estimation procedure must address. 9

11 empirical analysis. In particular, it is well known that persistence in a forecasting variable complicates the estimation of a predictive regression. In addition, any non-zero correlation between shocks to the predictor variable and shocks to the dependent variable induces a bias in the slope estimate of a predictive regression. Several papers including, among others, Stambaugh (1999), Nelson and Kim (1993), Lewellen (2004), Torous, Valkanov, and Yan (2005), and Paye and Timmermann (2006), address these and other issues in the estimation of a predictive regression. This research demonstrates that first-order asymptotic normality results are not reliable when applied to predictive regressions in small samples. While several improvements in estimating predictive regressions have been suggested, these improvements are difficult to apply in our setting because of the panel nature of our data as well as our relatively short sample period. In addition, we do not have an i.i.d. cross-sectional sampling scheme as is usually assumed in panel data studies. For these reasons, our estimation of predictive regressions will rely on a double resampling procedure which takes into account the bias inherent in the predictive regression s slope estimate, the overlapping nature of long horizon regressions, and the cross-sectional correlation in cap rates across metropolitan areas. This procedure, which extends Nelson and Kim (1993) s approach to a panel data setting, allows us to address these various estimation issues while directly accounting for our small sample size by relying on a bootstrap methodology. An alternative approach to deal with the persistence of a predictor variable in a predictive regression is to argue that this persistence reflects occasional structural breaks in the series (Paye and Timmermann (2006) and Lettau and Van Nieuwerburgh (2008)). For example, Lettau and Van Nieuwerburgh (2008) identify breaks in the aggregate stock market s dividend yield series and demonstrate that accounting for these breaks has an important effect on the stochastic behavior of the series and, more importantly, on predictability tests. We also test for structural breaks in our cap rate series by property type using Perron s (1989) sup-f test for breaks with unknown location. We consider both our main 1994:Q2 to 2003:Q1 sample period as well as the extended 1985:Q4 to 2002:Q4 sample period. Only in the case of office properties over the extended sample period do we find evidence of a statistically significant break in cap rates. This break, identified in 1992, corresponds to the well documented collapse in the U.S. office property market in the early 1990s. 16 The top panel of Figure 1 compares cap rates of commercial real estate on a national basis (solid line) with the dividend-price ratio of the stock market (dashed line) over the 1994:Q2 to 2003:Q1 sample period. The national commercial real estate cap series is calculated as the simple average of each of the four property type s national cap rate defined as a population-weighted 16 The difference in average office cap rates before versus after the estimated break is +0.78%. This result is consistent with, for example, the 24% drop in the level of the NCREIF office index between 1991 and No other NCREIF property index experienced a similar drop around this time period. 10

12 average of the corresponding cap rates in the various metropolitan area. 17 The stock market s dividend-price ratio is measured by the dividend price ratio of the CRSP value-weighted index. The average national cap rate is 9.1% while the average dividend-price ratio of the stock market over the same period is 1.7%. The bottom panel shows the same series on a biannual basis for the 1985:Q4 to 2002:Q4 sample period, where the dividend-price ratio series is break-adjusted in 1992 following the approach of Lettau and Van Nieuwerburgh (2008). 18 As can be seen, the stochastic properties of the two series are remarkably similar. In fact, the correlation between the stock market s dividend-price ratio and the national cap rate is for the 1994:Q2 to 2003:Q1 period and during the 1985:Q4 to 2002:Q4 period. This high correlation suggests that, at least at the aggregate level, the stock market and the commercial real estate market are influenced by common factors. To investigate their cross-sectional properties, Figure 2 plots the cross-sectional distribution of cap rates for each of the four property types. The figure displays mean cap rates as well as their 5 th and 95 th percentiles across the fifty-three metropolitan areas. As can be seen, cap rates exhibit considerable cross-sectional variation. For example, the average cross-sectional standard deviation in cap rates for offices is 0.58%, which is more than one and a half times the corresponding average time-series standard deviation of 0.37%. The average difference between the 5 th and 95 th percentiles of these distributions provides another measure of the cross-sectional variation in cap rates. In the case of apartments, the average difference between the 5 th and 95 th percentiles is 1.8% which is large when we recall that the average cap rate for apartments across metropolitan areas is 8.9%. This average difference is the largest for offices at 2.1%. Finally, we can also measure the cross-sectional dispersion in cap rates by looking at the average R 2 from regressing the individual cap rates on the national average for each corresponding property type. Consistent with the previous findings, the average R 2 is lowest for offices at 0.26 while increasing to 0.36 and 0.46 for industrial and retail properties, respectively. Even in the case of apartments, which display the highest degree of cross-sectional dispersion, the average R 2 is only To the extent that this considerable dispersion in cap rates reflects differences in expectations about future returns or rent growth rates across metropolitan areas, it represents valuable crosssectional information which we can exploit to improve the results of our predictability tests. 2.3 Expected Returns and Expected Growth in Rents Guided by these empirical properties, we now impose additional structure on our specification of commercial property returns and rent growth processes. The resultant framework allows us to not 17 Calculating the national cap rate as an average of equally weighted series gives a very similar picture. 18 The national commercial real estate cap series does not exhibit any statistically significant breaks. 11

13 only investigate in more detail the economic properties of our commercial real estate data but also to link the reduced-form predictive regressions to an underlying structural model. This approach is used by Lettau and Van Nieuwerburgh (2008) to model aggregate stock market returns. In particular, we assume r t+1 = r + x t + ξt+1 r (9) h t+1 = g + τx t + y t + ξt+1 h (10) where ξt+1 r represents an unexpected shock to commercial real estate returns and ξh t+1 unexpected shock to rent growth with E t (ξt+1 r ) = E t(ξt+1 h ) = 0. Taking expectations, gives is an E t r t+1 E t h t+1 = r + x t = g + τx t + y t where the variables x t and y t capture time variation in expected returns and expected rent growth, respectively. Notice that x t also enters into the equation describing expected rent growth. This will allow us to investigate whether expected returns and expected rent growth are correlated in the commercial real estate market. We model the variations x t and y t as mean-zero first order autoregressive processes: x t+1 = φx t + ξ x t+1 (11) y t+1 = φy t + ξ y t+1 (12) where ξt+1 x and ξy t+1 are mean zero innovations in expected returns and expected rent growth, respectively. 19 The system of equations (9)-(12) represents the structural model underlying the reduced-form predictive regressions. The four structural shocks of the system will be collected in the vector ξ t+1 = [ξ r t+1, ξh t+1, ξx t+1, ξy t+1 ] and the variances of these shocks are denoted by σ2 ξ r, σ2 ξ h, σ 2 ξ x, and σξ 2 y, respectively. The covariance structure of the structural model s shocks must be consistent with the loglinearized cap rate formula given by expression (2). In particular, Campbell s (1991) variance decomposition implies that the structural shocks must satisfy: ξ r t+1 = ρ [ (τ 1)ξ x 1 ρφ t+1 + ξ y ] t+1 + ξ h t+1 (13) 19 For simplicity, we assume that the autoregressive coefficients of the x and y processes are the same and are constant across the different metropolitan areas. While introducing area-specific φ values is an appealing extension, it would require the estimation of a much larger number of parameters which is not possible given our limited data. The modest cross-sectional variation in the cap rate s AR(1) coefficient across areas (not reported) suggests that such heterogeneity is not likely to be a first-order effect. 12

14 which guarantees identification of the structural model under the assumptions used to derive expression (2). Since expression (13) imposes one restriction among the four structural shocks, we must characterize the covariance structure of three of these shocks. To identify x t+1 and y t+1, we assume that their respective shocks are uncorrelated at all leads and lags, Cov(ξ y t+1, ξx t+j ) = 0, j. In addition, we assume that Cov(ξt+1 h, ξx t+j ) = 0 j, Cov(ξh t+1, ξy t+j ) = 0 j 1, and Cov(ξt+1 h, ξy t+1 ) = ϑ. Imposing these assumptions on the covariance structure of the three unique shocks in ξ t+1 allow us to identify τ as the impact of the expected return component x t+1 on rent growth conditional on y t+1. That is, τ captures any co-movement between expected returns and rent growth. For τ = 0, the time variation in expected returns does not influence rent growth, while for τ = 1, they move one for one. From these identifying assumptions, we can see that the variable y t represents fluctuations in expected rent growth that are orthogonal to expected returns. Using expressions (11) and (12), the cap rate can now be written as: cap t = r g 1 ρ ρφ [x t(1 τ) y t ]. (14) The first term in expression (14) reflects the difference between the unconditional expected return and the unconditional rent growth rate. The second term captures the influence of time-varying fluctuations in expected returns and rent growth rates on the cap rate. In particular, large deviations from the unconditional expected return (large x t ) or more persistent deviations (large φ) imply higher cap rates. Also, fluctuations in expected rent growth that are orthogonal to expected returns (y t ) are negatively correlated with cap rates. Finally, the autoregressive structure imposed on expected returns and expected rent growth implies that the cap rate itself follows an AR(1) process with autoregressive coefficient φ (see the proof in Appendix B). We can now explicitly relate the parameters of the reduced-form predictive regressions in expressions (3)-(4) to the structural parameters of the data generating system, expressions (9)- (12). 20 In particular, we can express the one-period predictive regression slope coefficients as follows: β = λ = (1 ρφ) (1 τ) + υ 1 1 τ (1 ρφ) [τ(τ 1) + υ] (1 τ) 2 + υ (15) (16) where υ = σ 2 ξ y/σ2 ξ x captures the strength of the orthogonal shocks ξy in expected rent growth relative to the expected return shocks ξ x. 20 All proofs are provided in Appendix B. 13

15 This result makes it clear that it is the combined effect of the structural parameters τ and υ which determines the sign and magnitude of the one-period predictive regression slope coefficients. 21 To better see this, the top two panels of Figure 3 plot the predictive coefficients β and λ for values of τ ranging between 1 and 1 given three empirically relevant υ values. The case τ = 0 corresponds to the common assumption made in the asset pricing literature (e.g., Campbell and Shiller (1988a), Cochrane (2008)) that expected returns do not affect cash flow growth rates. In this case, we see from expression (14) that the cap rate will be correlated with expected returns and expected rent growth rates. The one-period predictive slope coefficient from the return forecasting regression, expression (15), will be positive for τ = 0 while the corresponding coefficient from the rent growth forecasting equation, expression (16), will be negative. By contrast, for τ = 1, expected rental growth rates move one for one with expected returns and the cap rate in expression (14) will be unable to detect fluctuations in expected returns, β = 0, because the variation in expected returns will be exactly offset by corresponding fluctuations in expected rent growth rates. This can also be seen from expression (15) where β = 0 for τ = 1. In this case, however, λ remains negative because the cap rate is still negatively related to the future growth in rents, expression (14). 22 The link between return predictability and cash flow predictability, discussed recently by, among others, Cochrane (2008) and Lettau and Van Nieuwerburgh (2008), is also evident in the top two panels of Figure 3. For each value of υ, we see that as return predictability increases from τ = 0 through approximately τ = 0.5, the rent growth rate predictability coefficient λ decreases in absolute value. The opposite holds for subsequent τ values. In other words, for given values of the structural parameters, we must either observe return predictability, or rent growth predictability, or both. This result is a restatement of the present value relation (13) which, as pointed out in the context of the aggregate stock market by Lettau and Van Nieuwerburgh (2008), can also be expressed as a restriction between the predictive regression coefficients β and λ: β λ = 1 ρφ. (17) This economic restriction also accounts for the similarity of the plots in the top two panels of Figure 3 as the difference between β and λ must equal a constant for all values of the underlying structural parameters τ and υ. The effects of the structural parameter υ on the reduced-form parameters are explored in 21 We do not discuss the roles of the parameters φ and ρ because the former specifies the dynamics of an exogenous variable while the latter is a log-linearization constant which depends on the average cap rate and is seen to display little variation across property types (Table 1). In addition, notice from expressions (15) and (16) that as long as expected returns are stationary ( φ < 1), the quantity (1 ρφ) simply acts as a positive scaling quantity. This implies that φ and ρ do not affect the signs of the slope coefficients. 22 While negative values of τ are theoretically possible and are displayed in Figure 3, we will see that the τ estimates for our commercial real estate data suggest that expected returns and expected rental growth rates are positively correlated. 14

16 the bottom two panels of Figure 3 for three empirically relevant τ values. Since υ is defined as the strength of the orthogonal shocks in expected rent growth relative to expected return shocks, it plays an important role in determining the magnitude of β as well as the magnitude and sign of λ. 23 In the return predictability regression, υ can be interpreted as a noise-to-signal ratio as its numerator σ 2 ξ y captures the variation in ξ y orthogonal to expected return fluctuations. As υ increases, the signal from time-varying expected returns is dominated by orthogonal fluctuations in expected rent growth and return predictability becomes difficult to detect. Consequently, β decreases towards zero. By contrast, in the rent growth predictability regression, υ plays exactly the opposite role. It can now be interpreted as a signal-to-noise ratio because σξ 2 y captures the signal in the regression predicting rent growth rates. Therefore, as υ increases, predictions in the rent growth regressions become sharper. Explicitly linking the underlying structural parameters to the predictive regression s reducedform coefficients has economic as well as econometric advantages. In particular, the constraint given by expression (17) provides a link between the reduced-form estimates and the underlying structural model, thus allowing us to identify the sources of predictability. 24 We can also exploit two important restrictions which follow from this analysis in our empirical work. First, imposing this constraint allows us to jointly estimate the one-period predictive regression coefficients β and λ. Second, since the one-period predictive regression coefficients β and λ are related to the corresponding long-horizon predictive regression coefficients β k and λ k by expression (8), we will impose these cross-equation restrictions at a given horizon as well as across horizons in estimating β and λ. Both of these restrictions will lead to efficiency gains which takes on additional importance given our limited sample period. From the system of expressions (9)-(13), we can also derive the proportion of the variance of returns and rent growth due to time-variation in their unobservable expectations. These statistics correspond, respectively, to the R 2 s obtained when regressing realized returns on E t r t+1 (denoted by R 2 Er ), and realized rent growth rates on E t h (denoted by R 2 E h ). Koijen and Van Binsbergen (2009) propose these measures in their investigation of stock market predictability. While the identification and estimation of their underlying structural model differs from ours, using similar R 2 -based statistics will allow us to compare the observed time variation in commercial real estate to that of common stock. 23 We restrict our attention to the economically relevant case of υ > 0. In the degenerate case υ = 0, the expected growth in rents is entirely driven by shocks to expected returns when there are no orthogonal shocks to expected rent growth. 24 It should be noted that the identification of the structural parameters relies on the assumption that agents are forming their expectations rationally. If this were not the case, the Campbell and Shiller (1988b) decomposition ceases to be a valid description of the underlying data generating process as it would suffer from an omitted variable bias. This, in turn, would bias the interpretation of the structural parameters, but not the results from estimating the predictive VAR, equations (3-5). 15

17 It is important to realize that the statistics R 2 Er and R2 E h are based on the structural relations posited to prevail in the commercial real estate market. Therefore, they may differ substantially from those calculated in simple predictive regressions, such as expressions (3) and (4). To see the difference, consider the extreme case where expected returns and expected rent growth rates are time-varying and highly correlated so that τ = 1. Under this assumption, the cap rate will be unable to forecast future returns despite the fact that expected returns are time-varying and the R 2 of the predictive regression (3) will be zero. However, R 2 Er will be non-zero in this case and will be able to properly measure the magnitude of time-variation in expected returns. 3 Empirical Results 3.1 Estimation Method We use a Generalized Method of Moments (GMM) procedure to estimate the long-horizon predictive regressions, expressions (6) and (7), by imposing both the present value restriction, expression (17), as well as the restriction between short-horizon and long-horizon coefficients given in expression (8). We estimate these regressions in-sample at forecast horizons of k =1, 4, 8 and 12 quarters and the moments we use are the standard OLS orthogonality conditions. Taken together, we have a total of eight equations (four horizons for each of the two regressions) and two unknowns (β and λ). We rely on the pooled sample of fifty-three metropolitan areas over the 1994:Q2 to 2003:Q1 sample period for each of the four commercial property types. It is important to emphasize that our pooled regressions differ from the time-series regressions used in the stock return predictability literature. Given the limited time period spanned by our data, this pooled approach has a number of advantages. Because we are primarily interested in long-horizon relations but, unfortunately, do not have a sufficiently long time series, the only statistically reliable means of exploring these relations is to rely on pooled data. Also, as previously demonstrated, there is considerable heterogeneity in returns, rent growth rates, and cap rates across metropolitan areas at any particular point in time. Therefore, tests based on the pooled regressions are likely to have higher power than tests based on time-series regressions in which the predictive variable has only a modest variance (Torous and Valkanov (2001)). Before presenting our results, we address a number of statistical issues concerning our pooled predictive regression framework. First, the overlap in long-horizon returns and rent growth rates must be explicitly taken into account. In our case, this overlap is particularly large relative to the sample size. In addition to inducing serial correlation in the residuals, this overlap induces persistence in the regressors and, as a result, alters their stochastic properties. While this problem 16

18 has been investigated in the context of time-series regressions, it is also likely to affect the smallsample properties of the estimators in our pooled regressions. Second, the predictors themselves are cross-sectionally correlated. By way of example, the median cross-sectional correlation of apartment cap rates in our sample is and is as high as (between Washington, DC and Philadelphia). Because we effectively have fewer than fifty-three independent cap rate observations at any particular point in time, a failure to account for this cross-sectional dependence will lead to inflated t-statistics. Finally, the cap rates are themselves persistent and their innovations are correlated with the return innovations. Under these circumstances, it is well known that, at least in small-samples, least squares estimators of a slope coefficient will be biased in time-series predictive regressions (Stambaugh (1999)). In pooled predictive regressions, the slope estimates will also exhibit this bias because they are effectively weighted averages of the biased slope estimates of the time-series predictive regressions for each metropolitan area. As a result of these statistical issues, traditional asymptotic methods are unlikely to provide reliable inference in our pooled predictive regression framework. As an alternative, we rely on a two-step resampling approach. In the first step, as is customary when estimating any predictive regression, we run time-series regressions for each of the fifty-three metropolitan areas in which one-period returns and rent growth rates are individually regressed on lagged cap rates while the cap rates themselves are regressed on lagged cap rates. The coefficients and residuals from these regressions are subsequently stored. For each area we then resample the return residuals, rent growth residuals and cap rate residuals jointly across time (without replacement, as in Nelson and Kim (1993)). The randomized return and rent growth residuals are used to create one-period returns and rent growth rates, respectively, under the null of no predictability. To generate cap rates, we use the corresponding coefficient estimates together with the resampled residuals from the cap rate autoregression. For each resampling i, we then form overlapping multi-period returns and rent growth rates from the resampled single-period series and obtain GMM estimates (ˆβ i, ˆλ i ). This first step is used to obtain estimates of the small sample bias of the slope coefficients as the contemporaneous correlations between the return or rent growth residuals and cap rate residuals in a given metropolitan area as well as across areas are preserved. Because these data are generated under the null, the average ˆβ across resamplings, denoted by β = I 1 I ˆβ i=1 i, estimates the bias in the pooled predictive regression. The bias-adjusted estimate of β is obtained as ˆβ adj = ˆβ β, where ˆβ is the biased estimate of β obtained from the pooled GMM estimation. The bias-adjusted estimate for the rent growth predictive regression, ˆλ adj, is similarly constructed. This procedure is in essence that suggested by Nelson and Kim (1993) but applied to a pooled regression. However, while this procedure captures the overlap in the multi-period returns, it does not address the cross-sectional dependence in cap rates. This then necessitates our second step. To account for the possibility that our predictability 17

19 results are driven by cross-sectional correlation in cap rates, we resample the cap rates across metropolitan areas at each point in time for each forecast horizon. 25 Then, for a given cross-section, we estimate the predictive regression using the two-stage GMM procedure thus obtaining T estimates ( β s, λ s ), where T is the sample size for which we can construct GMM estimates using all horizons and the superscript s denotes cross-sectional estimates from the resampled data. We repeat the entire resampling procedure 1,000 times which gives 1,000 T estimates ( β s, λ s ). We use these 1,000 T estimates to compute standard errors, denoted by se(β) and se(λ), respectively. This second step is very similar to a standard Fama and MacBeth (1973) regression with the exception that we are running the regressions with 1,000 replications of bootstrapped data rather than the original data. The standard errors from the double resampling procedure account for both time-series and cross-sectional dependence in the data. In the first resampling step, the overlapping nature of the regressors is explicitly taken into account, while in the subsequent resampling step, at each point in time the cap rates are drawn from their empirical distribution. The bias-adjusted double resampling t statistic, denoted by t DR, is computed as t DR = ˆβ adj /se(β) = (ˆβ β)/se(β), and analogously for λ. The 95 th and 99 th percentiles of the bootstrapped distributions of the t DR statistic are used to assess the statistical significance in our empirical analyses. For the sake of clarity, we report levels of significance next to the estimates (5% and 1%) rather than small-sample critical values because the latter are a function of the overlap as well as the cross-sectional cap rate correlations of a particular property type Predictive Regression Results adj adj Table 2 presents the bias-adjusted GMM estimates, ˆβ k and ˆλ k, as well as their corresponding t DR statistics for each of the four property types based on the 1994:Q2 to 2003:Q1 sample period. The slope coefficients at the k = 1 quarter horizon and their t DR statistics are obtained directly from the GMM procedure. The long-horizon coefficients for k 4 quarters are then calculated using expression (8) where the required φ value is estimated by substituting the GMM estimates in the present value constraint expression (17). For these long-horizon slope coefficient estimates, the 25 The bootstrap is carried out with replacement. We also attempted resampling without replacement and obtained very similar results. 26 A few additional points are worth mentioning about the our resampling procedure. First, the second bootstrap is necessary in order to take into account the cross-sectional correlation in cap rates. Without it, the standard errors would not be corrected for the cross-sectional dependence in cap rates. We verified that if we use only the Nelson and Kim (1993) randomization, we obtain t statistics very similar to the Newey and West (1987) results. Second, it is interesting to note that the pooled regression does not produce unbiased estimates. The reason is that given the cross-sectional correlation in cap rates and the fact that cap rate fluctuations and return shocks are correlated, the pooled regression effectively yields a weighted average of the biased estimates that would have been obtained in time series regressions for each metropolitan area. Third, the bias-adjusted GMM estimates will take into account the fact that the bias tends to be larger at longer horizons, where the sample size is smaller and the overlap is larger. 18

20 t DR statistics are then calculated using the delta method. Statistical significance at the 5% and 1% levels are denoted by superscripts a and b, respectively. For comparison with previous results, we also provide the R 2 statistics for OLS regressions of future returns on log cap rates (R 2 r) and future rent growth on log cap rates (R 2 h ), estimated separately. For all property types, we see that at the short horizon of k = 1 quarter, cap rates are positively correlated with future returns. Retail properties have the largest bias-adjusted slope coefficient, ˆβadj = 0.111, and office properties have the smallest, ˆβadj = For all property types except offices, the corresponding bias-corrected slope coefficients are statistically significant. Regardless of the property type, however, cap rates explain very little of return variability at the k = 1 quarter horizon as evidenced by the low R 2 r statistics reflecting the large variability in quarterly commercial property returns. ˆβ adj k As the return horizon lengthens, k 4 quarters, the bias-corrected slope coefficients increase in magnitude and provide reliable evidence of return predictability in the case of apartments, industrial and retail properties. The predictive regressions also explain an increasingly larger proportion of the variability of these property returns at longer horizons. In the case of retail properties, the bias-corrected slope estimate increases to ˆβ adj = at k = 12 quarters where the predictive regression can explain approximately 27% of the variability in returns. The explanatory power of the predictive regressions are somewhat smaller for apartments and industrial properties, explaining 15% of industrial property returns and 17% of apartment returns at the k = 12 quarter horizon. In contrast to the other property types, there does not appear to be reliable evidence of return predictability at any horizon for offices. The bias-corrected slope estimates ˆβ adj k for offices are much smaller in magnitude and are never statistically significant. For example, at k = 12 quarters, the slope coefficient is ˆβ adj = 0.247, which is approximately half of the slope coefficient for apartments (0.468) and approximately one-quarter of that for retail properties (0.966). The corresponding R 2 r statistics are also small for offices across all horizons, achieving a maximum of only 3.9% at a horizon of k = 8 quarters. Turning our attention to the rent growth results presented in Table 2, it can immediately be seen that there is no reliable evidence that cap rates can forecast the future growth in rents of apartments, industrial and retail properties. Across all horizons, the bias-corrected slope coefficients ˆλ adj are not significantly different from zero and the corresponding R 2 h statistics are small, between 0% and 3.1 %. By contrast, in the case of offices, the bias-corrected slope coefficients are negative and statistically significant. For example, the slope coefficient is ˆλ adj = with t DR = at a horizon of k = 4 quarters, increasing to ˆλ adj = with t DR = at a horizon of k = 12 quarters. However, the explanatory power of these regressions is small, the R 2 h statistics 19

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